Turns out I was completely wrong, and 3D printing is actually pretty friggin amazing! 3D printers *literally* create objects in 3 dimensions by solidifying layers of powder in specific patterns. The 2D patterns for each layer are obtained by using a computer to 'slice' the object into virtual layers - typically 10 layers to a millimeter. The platform that the powder rests on moves downward so that the layers are solidified exactly one on top of the other, thereby re-creating the original 3D object.

Here's a clip from Nat Geo Amazing that explains and demonstrates the process beautifully -